A Classical Approach of Newtonian Mechanics

3 MOTION IN 3 DIMENSIONS 3.12 Relative velocity

0

z

x

R

Figure 17: The parabolic trajectory of a projectile

Note that the range attains its maximum value,

v 2

Rmax =

(^0) , (3.49)
g
when θ = 45 ◦. In other words, neglecting air resistance, a projectile travels
furthest when it is launched into the air at 45 ◦ to the horizontal.
The maximum altitude h of the projectile is attained when vz = dz/dt = 0
(i.e., when the projectile has just stopped rising and is about to start falling). It
follows from Eq. (3.46) that the maximum altitude occurs at time t 0 = v 0 sin θ/g.
Hence,
h = z(t 0 ) =
Obviously, the largest value of h,
v^
2
2 g
v 2
sin^2 θ. (3.50)
hmax =
(^0) , (3.51)
2 g
is obtained when the projectile is launched vertically upwards (i.e., θ = 90 ◦).
1.25 Relative velocity
Suppose that, on a windy day, an airplane moves with constant velocity va with
respect to the air, and that the air moves with constant velocity u with respect
h
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